Solving Number Interchange Problems in Equations

The question asks us to find which pair of numbers, when interchanged in the given equation, makes the equation mathematically correct. The given equation is:

\(9 + 7 \times 5 – 18 \div 2 = 3 \times 4 – 10 + 45 \div 5\)

To check if the equation is correct, we need to evaluate both sides using the order of operations (BODMAS/PEMDAS).

Applying Order of Operations (BODMAS/PEMDAS)

BODMAS stands for Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction. PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. Both rules dictate the sequence of operations.

Evaluating the Original Equation

Let's evaluate the Left Hand Side (LHS) and Right Hand Side (RHS) of the original equation:

LHS: \(9 + 7 \times 5 – 18 \div 2\)

• First, perform Multiplication and Division (from left to right): \(7 \times 5 = 35\) and \(18 \div 2 = 9\)
• LHS becomes: \(9 + 35 – 9\)
• Next, perform Addition and Subtraction (from left to right): \(9 + 35 = 44\)
• LHS becomes: \(44 – 9 = 35\)

RHS: \(3 \times 4 – 10 + 45 \div 5\)

• First, perform Multiplication and Division (from left to right): \(3 \times 4 = 12\) and \(45 \div 5 = 9\)
• RHS becomes: \(12 – 10 + 9\)
• Next, perform Addition and Subtraction (from left to right): \(12 – 10 = 2\)
• RHS becomes: \(2 + 9 = 11\)

Since \(35 \neq 11\), the original equation is incorrect.

Testing the Options by Swapping Numbers

Now, let's test each given option by swapping the indicated numbers in the equation and re-evaluating both sides.

Option 1: Swapping 2 and 5

The equation becomes: \(9 + 7 \times 2 – 18 \div 5 = 3 \times 4 – 10 + 45 \div 2\)

LHS: \(9 + 7 \times 2 – 18 \div 5\)

• Multiply/Divide: \(7 \times 2 = 14\), \(18 \div 5 = 3.6\)
• LHS becomes: \(9 + 14 – 3.6\)
• Add/Subtract: \(9 + 14 = 23\), \(23 – 3.6 = 19.4\)

RHS: \(3 \times 4 – 10 + 45 \div 2\)

• Multiply/Divide: \(3 \times 4 = 12\), \(45 \div 2 = 22.5\)
• RHS becomes: \(12 – 10 + 22.5\)
• Add/Subtract: \(12 – 10 = 2\), \(2 + 22.5 = 24.5\)

Since \(19.4 \neq 24.5\), swapping 2 and 5 does not make the equation correct.

Option 2: Swapping 18 and 45

The equation becomes: \(9 + 7 \times 5 – 45 \div 2 = 3 \times 4 – 10 + 18 \div 5\)

LHS: \(9 + 7 \times 5 – 45 \div 2\)

• Multiply/Divide: \(7 \times 5 = 35\), \(45 \div 2 = 22.5\)
• LHS becomes: \(9 + 35 – 22.5\)
• Add/Subtract: \(9 + 35 = 44\), \(44 – 22.5 = 21.5\)

RHS: \(3 \times 4 – 10 + 18 \div 5\)

• Multiply/Divide: \(3 \times 4 = 12\), \(18 \div 5 = 3.6\)
• RHS becomes: \(12 – 10 + 3.6\)
• Add/Subtract: \(12 – 10 = 2\), \(2 + 3.6 = 5.6\)

Since \(21.5 \neq 5.6\), swapping 18 and 45 does not make the equation correct.

Option 3: Swapping 9 and 3

The equation becomes: \(3 + 7 \times 5 – 18 \div 2 = 9 \times 4 – 10 + 45 \div 5\)

LHS: \(3 + 7 \times 5 – 18 \div 2\)

• Multiply/Divide: \(7 \times 5 = 35\), \(18 \div 2 = 9\)
• LHS becomes: \(3 + 35 – 9\)
• Add/Subtract: \(3 + 35 = 38\), \(38 – 9 = 29\)

RHS: \(9 \times 4 – 10 + 45 \div 5\)

• Multiply/Divide: \(9 \times 4 = 36\), \(45 \div 5 = 9\)
• RHS becomes: \(36 – 10 + 9\)
• Add/Subtract: \(36 – 10 = 26\), \(26 + 9 = 35\)

Since \(29 \neq 35\), swapping 9 and 3 does not make the equation correct.

Option 4: Swapping 7 and 4

The equation becomes: \(9 + 4 \times 5 – 18 \div 2 = 3 \times 7 – 10 + 45 \div 5\)

LHS: \(9 + 4 \times 5 – 18 \div 2\)

• Multiply/Divide: \(4 \times 5 = 20\), \(18 \div 2 = 9\)
• LHS becomes: \(9 + 20 – 9\)
• Add/Subtract: \(9 + 20 = 29\), \(29 – 9 = 20\)

RHS: \(3 \times 7 – 10 + 45 \div 5\)

• Multiply/Divide: \(3 \times 7 = 21\), \(45 \div 5 = 9\)
• RHS becomes: \(21 – 10 + 9\)
• Add/Subtract: \(21 – 10 = 11\), \(11 + 9 = 20\)

Since \(20 = 20\), swapping 7 and 4 makes the equation correct.

Therefore, interchanging the numbers 7 and 4 makes the given equation correct.

Revision Table: Equation Evaluation with Swaps

Additional Information: The Importance of BODMAS/PEMDAS

The order of operations is crucial in evaluating mathematical expressions to ensure a single, correct result. Without a standard order, expressions could be interpreted in multiple ways, leading to different answers. The BODMAS or PEMDAS rule provides this standard order:

• B/P: Brackets/Parentheses - Evaluate expressions inside brackets first.
• O/E: Orders/Exponents - Evaluate powers and roots next.
• DM: Division and Multiplication - Perform these operations from left to right.
• AS: Addition and Subtraction - Perform these operations from left to right.

In this number interchange problem, correctly applying the BODMAS/PEMDAS rule to both sides of the equation after each swap is essential to determine if the equation becomes correct.